Supersymmetry, lattice fermions, independence complexes and cohomology theory

We analyze the quantum ground state structure of a specific model of itinerant, strongly interacting lattice fermions. The interactions are tuned to make the model supersymmetric. Due to this, quantum ground states are in one-to-one correspondence with cohomology classes of the so-called independence complex of the lattice. Our main result is a complete description of the cohomology, and thereby of the quantum ground states, for a two-dimensional square lattice with periodic boundary conditions. Our work builds on results by Jonsson, who determined the Euler characteristic (Witten index) via a correspondence with rhombus tilings of the plane. We prove a theorem, first conjectured by Fendley, which relates dimensions of the cohomology at grade n to the number of rhombus tilings with n rhombi.